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Neutron yields and energy spectra from the thick target Li(p,n) source
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1976 J. Phys. D: Appl. Phys. 9 15
(http://iopscience.iop.org/0022-3727/9/1/008)
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J. Phys. D: Appl. Phys., Vol. 9, 1976. Printed in Great Britain. 0 1976
Neutron yields and energy spectra from the thick target Li(p, n)
source
A I M Ritchie
Australian Atomic Energy Commission, Research Establishment, New Illawarra
Road, Lucas Heights 2232, NSW
Received 9 June 1975, in final form 4 August 1975
Abstract. Angle-dependent spectra of neutrons, emitted by a thick lithium target when
bombarded with protons in the energy range 1.881-3 MeV, have been evaluated from
experimental and theoretical values of the angular distribution of neutrons emitted in
the %(p, n) 'Be reaction. The variation of d2N/dEndQ with proton energy at different
angles of emission, and with angle of emission for different neutron energies, is presented
for the ground and excited state reactions. The angle-integrated neutron spectrum is
given for different incidence proton energies.
1. Introduction
The 7Li(p, n) 7Be reaction is a relatively prolific source of neutrons for proton bombarding energies in the region of 3 MeV. It is a useful source of neutrons for fast reactor
experiments, since the spectrum of neutrons emitted from a thick target bombarded with
3 MeV protons covers the range 0-1.3 MeV, with the bulk of neutrons in the range
50 keV-1.0 MeV. It has the added advantage compared, say, with the 9Be(d, n) 1OB
reaction, that the neutron energy spectrum is closely controlled by the energy of the
incident protons.
Although it is a relatively straightforward process to use experimentally measured
differential cross sections (d2o/dQdE,) to present the number of neutrons emitted at
various angles as a function of proton bombarding energy (Theobald et al 1971), the
information is not in a form suitable for reactor calculations. Such calculations require
the angular distribution of neutrons emitted from the source as a function of neutron
energy. Usually, this information is required in the form of Legendre moments of the
neutron energy spectrum, the commonest requirement being the zeroth moment, which
is just the angle-averaged or total spectrum.
In the present paper, both experimental and theoretical data for the ground and
excited state 7Li(p, n) 'Be reactions have been used to present the number of neutrons
emitted:
(i) as a function of proton energy for various angles of neutron emission, and
(ii) as a function of angle for various neutron energies for protons in the energy
range 1.881-3 MeV.
The data have been further processed to give the total neutron spectrum for a thick
target at different proton bombarding energies. Some indication is also given of the
extent to which the total neutron spectrum can be tailored by varying the target thickness.
15
16
A I M Ritchie
2. Formalism for calculating the neutron energy spectrum
The number of neutrons emitted per second into solid angle dQ by an element dx of a
thick target bombarded by a proton beam of i pA is
dN= igD (da(Ep)/dQ) dx dQ,
where Ep is the proton energy at depth x from the surface of the target, g is the number
of protons per PA, D is the atomic density of 7Li, and da/dQ is the differential cross
section in the laboratory frame.
Since there is a well defined relationship between the proton energy and depth of
penetration into the target, and between the proton energy Ep and the energy E n of the
neutron emitted into the solid angle Q, we can express dx as
dx=(dx/dEp) ldEp/dEnl dEn.
The number of neutrons emitted can be usefully expressed in terms of the differential
cross section referred to the centre of mass frame, since measurements and theoretical
expressions for the differential cross section are most frequently quoted in this frame of
reference. The differential neutron energy spectrum then takes the form
We can consider separately the contributions of the ground and excited state reactions
to the total neutron energy spectrum. For each, there are analytical expressions for
dQ2'/dQand dEp/dEn, which are most succinctly expressed in the form (Winter and
Schmid 1968)
(2"
+
l/y2- 1 p*.2
(5)
where ,U is the cosine of the angle of neutron emission in the laboratory frame; mn, mp,
my are the mass number of the neutron, proton and 7Li respectively and E t h is the threshold energy (1,881 MeV for the ground state reaction, 2.378 MeV for the excited state
reaction). In the same notation, the relationship between Ep, E n and p takes the form
When 1 I
yI
CO, the neutron energy is a double-valued function of the proton energy,
the two values being given by the positive and negative values of (. Similarly, dQ'/dQ
and dEp/dEn are double-valued functions of proton energy. In equations (2) and (3),
the positive value of is used to evaluate dQ'jdQ and dEp/dEn, associated with the
larger value of E n in equation (6).
The variation of da/dQ2'with Ep is not, in general, a simple function of Ep and has
to be evaluated from experimental results. On the other hand, an analytical expression
Neutron yields and energy spectra f r o m the thick target Li(p, n) source 17
(Livingston and Bethe 1937) does exist for the slowing-down power dEp/dx, characterized by I, the mean excitation potential. The choice of I and the agreement between
theoretical and experimental values of the proton energy loss are discussed below.
3. Discussion of data used in evaluating da/dQ’ and dE,,/dx
There is surprisingly little information on the neutron angular distribution from the
7Li (p, n) 7Be reaction as a function of proton energy in the range 1*88-3 MeV. Bergstrom
et al (1967) have carried out angular distribution measurements in the range
1.928-2.361 MeV, while Buccino et al (1964) have measured angular distributions of
neutrons from both the ground and excited states in the range 2-48-2.977 MeV. Bevington
et al(l961) have also measured the angular cross section for the ground and excited states
at a limited number of points in the present energy range. In the region near threshold,
the ground state reaction is isotropic in the centre of mass system, and the energy dependence can be described by a single resonance Breit-Wigner formulation (Macklin and
Gibbons 1958).
Macklin and Gibbons have shown that the energy level near threshold is broad and
that the Breit-Wigner formula reduces to
where
x = Fn/Fp=the ratio of the neutron to proton widths
and FOand Go are the regular and irregular Coulomb functions. Now (Fo2+ Go2) varies
only slowly over the limited energy range of interest and a good fit (figure la) to the
total cross section is obtained by using
This fit is better than that obtained using yn2/yp2= 5.2 as derived by Macklin and Gibbons
(see figure la) and suggests that a better value for yn2/yp2would be 3.5. The differential
cross section was evaluated in the range 1.881-1.928 MeV using equation (8) in equation
(7).
The angular distribution data of Bergstrom et a1 were integrated over solid angle
and compared with the total cross section (figure lb) given in the review of Gibbons
and Newson (1960). The agreement is good, but it should be noted that it is necessary
to correct two typographical errors that appear in the paper by Bergstrom et al-at
Ep=2*161, A o should be 35 rather than 30, and at Ep=2*231, Ao should be 0 rather
than 10.
The angular distribution data of Buccino et al for the excited state indicate that near
threshold, the angular distribution in the centre of mass is nearly isotropic and the cross
section can again be described by a single Breit-Wigner resonance. It is also clear from
3
18
A I M Ritchie
/
"
I
/
600-(')
500
v
-
400
I
1.88 1.90 1.92 L94 1.96
2.0 2.2 2.4 2.6 2.8
Ep (MeV)
J-
2.4 2.6 2.8 3.0
Figure 1. Comparison of angle-integrated differential cross section data with total cross
sections: (a), ground state near threshold, -Macklin and Gibbon experiment (1958)
-- ~ = 4 * 7 (1-Eth/Ep)112,
2
--.- X z 5 . 2 (Fo2+Go2)(1-Eth/Ep)'I2; (b), ground State
1.881 <ED<2*977MeV, 0 Bergstrom et al(1967), A Buccino et a1 (1964), -Gibbons
MeV, A Buccino et al,
and Newson (1960 atotai); (c), Excited state 2*3784Epd2*977
-- Single resonance theory, -Bevington et aZ(1961 ototal).
the data that the angular distribution becomes increasingly anisotropic as EDincreases,
but in the present calculations, do/dQ' for the excited state has been evaluated over the
whole range from threshold at 2,378-3 MeV, using the single resonance expression
and the parameters given by Buccino et al. The angular distribution predicted by the
simple expression is in reasonably good agreement with the results of Bevington et a1
except at the higher energies where the maximum deviation is about 30%. Since the
excited state cross section is only about 10% that of the ground state, the error in the
neutron energy spectrum introduced by using this approximation is expected to be small.
The single resonance approximation was used rather than the experimental values of
Buccino et al since these data, when integrated over solid angle, gave poor agreement
(see figure I C ) with the total cross section (Bevington et all961, Batchelor and Morrison
1955), indicating some uncertainty in the method used by Buccino et a1 to normalize
their angular distributions to the total cross section.
The angular distribution data of Buccino et al for the ground state reaction were also
integrated over solid angle and compared with the total cross section of Gibbons and
Newson (1960) from which the excited state contribution had been subtracted. The
Buccino et a1 cross section has the same energy dependence, but gives better agreement
with the Gibbons and Newson curve when reduced by 6 7;. This correction was applied
to the Buccino et a1 data in calculating the neutron energy spectra.
The total cross section curve of Gibbons and Newson has a relative accuracy of
about +_ 1 % and an absolute accuracy of about 14%. Bergstrom et a1 quote accuracies
of 10% for their angular distribution data, and Buccino et a1 quote If: 10% up to 2.7 MeV
Neutron yields and energy spectra from the thick target Li(p,n) source 19
and i- 15% above 2.7 MeV. It would therefore seem reasonable to expect the present
derived angular neutron spectra to be accurate to about 10-15 %.
The expression for the stopping power derived by Livingston and Bethe takes the
form
__-2ne4z2DZ mp (log 4Epme log (1 - 8 3 - 8 2 ) .
dEp
dx
Ep
me
IMP
Since in the present energy regime, p, the ratio of the proton velocity to the velocity of
light is very small, the stopping power can be characterized by one parameter, I, the
mean excitation potential. Aron et aZ(l949) have used I= 34.5 eV, while Whaling (1958)
has assigned I = 6 0 eV on the basis of some accurate energy loss measurements in the
range 0.1 < EpI 1.0 MeV. These two expressions give absolute values of the angular
neutron spectra differing by about 12%, but relative variations change by only 1-2%.
Scott (1 971), who found excellent agreement between measured thick-target total yields
and yields calculated using Whaling’s value for the slowing-down power, estimated that
in the energy range 3-10 MeV the Whaling value had a systematic error of less than
6.5 %.
Whaling’s value for I has been used in the present calculations and the slowing-down
power for lithium is taken as
E
= D-1 dEp/dx= 7.18 x 10-16 [4.69-log(3/Ep)] Ep-f eV cm2,
where Ep is in MeV. Based on the accuracy of the experimental data and extrapolating
Scott’s findings below 3 MeV, the calculated angular spectra should have a relative
accuracy of 10-15 %, with an additional uncertainty of & 7 % in the absolute yields.
4. Calculated angular and total neutron spectra
The differential neutron yield at various angles and at the proton energies at which angular
distributions were measured, was evaluated by direct substitution of the experimental
values of du/dQ’ in equation (1). The differential neutron yields from the ground state
reaction near threshold and from the excited state were calculated using the single
Breit-Wigner resonance expression for du/dQ‘ and the parameters for the resonances
indicated above. Figures 2 and 3 show the differential yields for the ground and excited
states respectively.
It should be noted that the differential neutron yields in these figures are plotted as
functions of incident proton energy rather than functions of observed neutron energy
which is the more normal abscissa to use in plotting a neutron spectrum. However, the
mode of presentation used is probably more generally useful since proton energy rather
than neutron energy is the parameter under direct control of the experimenter in most
practical applications.
The double-valued behaviour of the yield at angles less than 90” and for proton
energies less than 1.92 MeV in figure 2 corresponds to the double-valued behaviour of
the neutron energy as a function of proton energy in this range. The lower part of the
curve corresponds to the lower neutron energies, all of which lie below 30 keV. For ease
of presentation, the lower energy branch has been omitted from the neutron energy
scales of figures 2 and 3, and the double-valued portion of the yield curve has been
omitted from figure 3. It should be stressed that the neutron energy scales of figures 2
and 3 are intended only as a guide to the neutron energy associated with each point of
20
A I M Ritchie
2.0
2.2
2.4
2.6
Proton energy (MeV)
2.8
Figure 2. Differential neutron yield from a thick target for the ground state as a function
of proton energy for various angles. (a) 30"; A 90"; e 150"; (b) 0",A 60°, 0 120".
+
+
Neutron yields and energy spectra f r o m the thick target L i ( p , 12) source 21
Figure 3. Differential neutron yield from a thick target for the excited state as a function
of proton energy for various angles: (U)-0", - - - - 60°,.-.-.120"; (b)-30",
----- 90°,
-.*-
150".
22
A I M Ritchie
the neutron differential yield curves plotted. In all cases, the yields correspond to the
isotopic abundance of 7Li in natural lithium.
It is a slightly more involved process to calculate d2N/dEndQ as a function of p
for a given En. Particular values of p and En define a particular value of Ep according
to the expression
+
Ep-En ( -my
+ 2 nzn
my -mP
p2
mp mn
Eth -
En2 [(my-mp)(my+mn) mn mp+mn2 mp2 p21
imn - mp)(my-mp)
+ EnEth nznmpmy (my(my
+mn)
For resulting values of Ep close to threshold for the ground state reaction, and for all
values of E p for the excited state reaction, the angular cross section as a function of p
can be found by straightforward substitution into the analytical expressions for du/dQ2'.
For resulting values of Ep lying within the range covered by the experiments of Bergstrom et a1 and Buccino et al, the value of du/dQ' at the required value of ED was found
,
_
I
_
_
I
-I.0-0*8-0.6-0.4-0*2 0.0 0.2 0.4 0.6 0.8 1.0
c
Figure 4. Differential neutron yield from a thick target for the ground state as a function
of angle for various neutron energies in keV: (a) -50, - - - - 100, - - 200,
--300; (b)-400, - - - 500, - - - 600,--700, (c) -800, - - - - 900,
---1000, --1100.
-
'
20.0
1
I
,
I
I
I
,
I
(0)
16.0-
f
\
8.0-
2w
\
\
'\
-5
5-
..............
.........
5
g
\
...-.
a
a
\
.....
12.0-'---.,_
Y)
a
\
<
.\'
-- --.---_
.................................
................_
:
4.0:
&
I!
'3225.0
I
2
I
3
I
i
g3
t
8
(bl,
5
20.015.0-
10.0-
5.0,
-1.0-0,8-0,6-0,4-0,2
,
I
C
,
,
0 0.2 0.4 0.6 0.8 1.0
by interpolation. The interpolation technique used in practically all cases was to fit a
third-order polynomial through the angular distribution measurements at the two energy
points immediately below, and the two immediately above the one required. When the
Epvalue lay between the lowest two points used by Bergstrom et aZ(1.928 and 2.031 MeV),
the angular distribution corresponding to Ep= 1.9 MeV was evaluated using the BreitWigner resonance expression, and the four-point interpolation method used as before.
For Epvalues lying between the highest and second-highest of the values used by Buccino
et aZ, a simple linear interpolation method was used.
, for various En is shown in figures 4 and 5 for the
The variation of d'N/dEndQ with U
ground and excited states respectively. It should be noted that the backward angle cut-off
in these figures corresponds to a proton energy of 2.977 MeV, the highest energy used
by Buccino et al.
The energy spectrum of neutrons emitted into all possible angles can be derived by
integrating the differential cross section over p. This was done numerically using a
, the integration for a chosen neutron
simple trapezoidal rule. The lower limit, p ~ on
incident on the thick target, is given by
energy and given maximum proton energy Epmax
As En increases, p~ approaches 1 for a given Epmax.Physically, this follows from the
fact that the maximum neutron energy is achieved by protons of the maximum energy
24
A I M Ritchie
interacting with lithium at the surface of the target and emitting neutrons in the forward
direction. Neutrons of slightly lower energies can be produced either at the target
surface, but emitted at some angle greater than zero, or at some point below the surface
and emitted in the forward direction. Neutrons of even lower energy can be produced
over a wider angular range until, at sufficiently low neutron energies, neutrons are
produced at all angles.
The neutron energy spectra corresponding to a thick target and maximum incident
proton energies of 2.1, 2.4 and 2-8 MeV are shown in figure 6. Table 1 shows the contribution of the ground state and excited state reactions to the total neutron energy spectra
for the two incident proton energies above the excited state threshold.
I
,
-,
,
1
Neutron energy IkeVl
Figure 6. Angle-integrated neutron spectra for thick targets and different incident
proton energies: -2.1 ; - - - 2.4; - - - 2%
It is also possible, by changing the target thickness and by judicious choice of the
proton bombarding energy, to produce angle integrated spectra which peak at different
neutron energies in the range 50-700 keV, a range of considerable interest in fast reactor
neutronics. The extent to which this is possible can be judged from figure 7. The maximum proton energy determines the number of high-energy neutrons, while the yield of
low-energy neutrons can be reduced by choosing a thin target which eliminates the
reactions involving low-energy protons from which come the bulk of low-energy neutrons.
Choosing a thin target, in fact, introduces a low-energy proton limit E p m i n which in turn
introduces an upper limit to the integration over angle for some neutron energies.
This corresponds physically to the fact that when Ep is greater than the value for which
the neutron energy is a double-valued function of proton energy, there is a minimum
possible neutron energy for a given E p m i n , and this occurs for neutrons emitted backwards ( p = - 1) in the laboratory frame of reference. Figure 7 shows the angle-integrated
neutron spectra for incident proton energies of 2.1, 2.4 and 2.8 MeV when the target
thickness is 220, 200 and 500 keV, respectively. For the two higher proton energies, the
low-energy neutron tail is due to the excited state reaction.
Neutron yields and energy spectra f r o m the thick target Li(p, n) source 25
Table 1. Angle-integrated thick target neutron spectrum from ground and excited state
at proton energies of 2.4 and 2.8 MeV.
Neutron
energy
20
50
75
100
200
300
400
500
600
700
800
900
1000
Neutron yield n eV-1 FA-1 s-1
Epmhx
= 2.4 MeV
Epmax
= 2.8 MeV
Ground
state ( x 10-2)
Excited
state ( x 10-2)
Ground
state ( x 10-2)
Excited
state ( x 10-2)
9.670
14.586
16.411
17.326
14,387
12.438
13,457
12.036
5.583
0.025
0.029
9.670
14.586
16.411
17.326
14.576
15,342
19.235
19,430
15.416
11.377
8.256
5.548
2.781
0,219
0.372
0.484
0.594
0.654
0335
0.390
0.217
0,014
0,019
0.004
-
-
Neutron energy (keV1
Figure 7. Angle-intergrated neutron spectra for different target thickness and different
incident proton energies: - - - Epc2.8, AEp=O*5; - Epz2.4, AED=0*2; - - Epz2.1, AEpzO.22.
-
The calculation of neutron spectra and angular neutron spectra presented in this
paper were carried out using a Fortran code which operates on an IBM 360/50. The
data presented was intended to indicate the range of spectra possible using a Li (p, n)
thick target source and it is clear that a potential user of this source could well be interested in a different range of angles, neutron energies, incident proton energies and target
thicknesses from those presented here. Details of the computer code are available from
the author.
26
A I M Ritchie
Acknowledgments
The assistance of Miss J Sims and Mr L Sullivan in the preparation of this work is
greatly appreciated.
References
Aron W A, Hoffman B G and Williams F C 1949 AECU 663
Batchelor R and Morrison G C 1955 Proc. Phys. Soc. A 68 1081
Bergstrom A, Schwarz S , Stromberg L G and Wallin L 1967 Ark. Fys. 34 153
Bevington P R, Rolland W W and Lewis H W 1961Phys. Rev. 121 871
Buccino SG, Hollandsworth C E and Bevington P R 1964, Nucl. Phys. 53 375
Gibbons J H and Newson H W 1960 Fast Neutron Physics Vol 1, ed J B Marion and J L Fowler
(New York: Interscience) p 133
Livingston M S and Bethe H 1937 Rev. Mod. Phys. 9 263
Macklin R L and Gibbons J H 1958 Phys. Rev. 109 105
Scott M C 1971 J. Nucl. Energy 25 405
Theobald J P , Migneco E and Cervini C 1971 Nucl. Znstrum. Meth. 95 1
Whaling W 1958 Hundb. Phys., Vol 34 ed S Fliigge (Berlin: Springer-Verlag) p 193
Winter J and Schmid H 1968 EUR 3908e.